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Cite This: Nano Lett. 2019, 19, 7028−7034
Quantum Hall Effect Measurement of Spin−Orbit Coupling Strengths in Ultraclean Bilayer Graphene/WSe2 Heterostructures Dongying Wang,† Shi Che,† Guixin Cao,† Rui Lyu,‡ Kenji Watanabe,§ Takashi Taniguchi,§ Chun Ning Lau,† and Marc Bockrath*,† †
Department of Physics, The Ohio State University, Columbus, Ohio 43210, United States Department of Physics and Astronomy, University of California, Riverside, California 92521, United States § National Institute for Materials Science, Namiki Tsukuba Ibaraki 305-0044 Japan Downloaded via UNIV OF NEW ENGLAND on October 22, 2019 at 20:27:37 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
‡
S Supporting Information *
ABSTRACT: We study proximity-induced spin−orbit coupling (SOC) in bilayer graphene/few-layer WSe2 heterostructure devices. Contact mode atomic force microscopy (AFM) cleaning yields ultraclean interfaces and high-mobility devices. In a perpendicular magnetic field, we measure the quantum Hall effect to determine the Landau level structure in the presence of out-of-plane Ising and in-plane Rashba SOC. A distinct Landau level crossing pattern emerges when tuning the charge density and displacement field independently with dual gates, originating from a layer-selective SOC proximity effect. Analyzing the Landau level crossings and measured inter-Landau level energy gaps yields the proximity-induced SOC energy scale. The Ising SOC is ∼2.2 meV, 100 times higher than the intrinsic SOC in graphene, whereas its sign is consistent with theories predicting a dependence of SOC on interlayer twist angle. The Rashba SOC is ∼15 meV. Finally, we infer the magnetic field dependence of the inter-Landau level Coulomb interactions. These ultraclean bilayer graphene/WSe2 heterostructures provide a high mobility system with the potential to realize novel topological electronic states and manipulate spins in nanostructures. KEYWORDS: Graphene, transition-metal dichalcogenide, spin−orbit coupling, quantum transport fforts to control spin and electronic ground state topology in devices has driven an intensive study of spin−orbit coupling in materials.1,2 Graphene is a two-dimensional (2D) material with excellent electronic properties but small spin− orbit coupling (SOC),3−6 motivating efforts to increase it.7−21 For example, adatoms such as hydrogen or transition metals increase graphene’s SOC,7,8 however these add disorder and lower charge mobility.7 In another approach, coupling graphene to 2D materials with strong SOC such as transition metal dichalcogenides also increases graphene’s SOC,9−16,18,19,21 whereas the interfacing of two crystalline systems adds less intrinsic disorder. Such studies have shown weak antilocalization signatures of spin−orbit coupling9,10,13−16 as well as spin Hall signatures.12 Shubnikov-de Haas oscillations were used to infer a dominant Rashba SOC of magnitude ∼10−15 meV.9 However, electrical transport studies in high mobility devices on the quantum Hall effect are limited which can act as a precise simultaneous probe of both
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© 2019 American Chemical Society
Ising and Rashba SOC in graphene and potentially realize new topological ground states.19,20,22 Moreover, although recent progress has been made by a combination of capacitance and some transport measurements,22 inter-Landau level energy gaps have not been measured, nor the Rashba SOC or magnetic field dependence of the inter-Landau level Coulomb interactions. Here we report coupling bilayer graphene (BLG) via layer stacking to WSe2, a 2D semiconductor with strong SOC, in hexagonal BN (hBN) encapsulated devices. We obtain high mobility devices (up to ∼110 000 cm2 V−1 s−1) by squeezing contaminants away from device areas using contact AFM.23 Our devices are dual gated, allowing independent tuning of the Received: June 15, 2019 Revised: September 15, 2019 Published: September 17, 2019 7028
DOI: 10.1021/acs.nanolett.9b02445 Nano Lett. 2019, 19, 7028−7034
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Figure 1. Device structure and SOC modified LL properties. (a) Main panel: Rxx versus VBG at T = 1.5 K with VTG = 0. Left inset: Optical image of a BLG/WSe2 device. The dashed outline indicates the top gated region before metalization. Scale bar is 2 μm. Right inset: Schematic diagram of the layer stack. (b) Rxx versus VBG and magnetic field B showing a Landau Fan pattern. Quantum Hall states at all integer filling factors of the Landau levels are clearly visible due to the full degeneracy lifting. (c) Top panel: Rxx versus n at D = 0. Bottom panel: Color plot of Rxx versus n and D at B = 10 T. (d) Vertical line cuts passing Landau level crossing points along the dashed lines in panel c. Each curve represents a fixed filling factor for ν = 5, 6, and 7.
charge density n and perpendicular displacement field D. At low temperatures, under a perpendicular magnetic field B, we observe the quantum Hall effect (QHE) with all degeneracies lifted for D = 0, resulting from inversion symmetry breaking by the WSe2 layer contacting the graphene. Applying a nonzero D yields a finite interlayer potential difference, enabling us to observe Landau level (LL) crossings24−28 that differ in D values from isolated BLG. We obtained high quality data from two devices, D1 and D2. To understand this behavior, we compare our results to a single-particle theory that includes an out-of-plane Ising SOC λ as well as an in-plane Rashba SOC λR.20 For the nearly degenerate zero energy LL octet, most crossings deviate from the single-particle picture due to Coulomb interactions except for the ν = ±3 crossings.20,22,29 From these we extract λ ∼ −2.2 ± 0.1 meV, where the sign indicates the direction of the effective out-of-plane magnetic field from the SOC in the bilayer graphene at the K+ point (see SI for additional details) of the Brillouin zone (with the opposite direction at the K− point),20 consistent with recent theoretical work predicting that λ can be negative depending on the graphene/WSe2 twist angle.30 To gain further insight into the device behavior, we measure the inter-LL gaps using temperature-dependent transport measurements. These gaps are potentially affected by both the LL width as well as Coulomb interactions. We therefore study the rate of gap closure with the displacement field which is independent of these within a Hartree−Fock model. Measuring these rates for several gaps, we find the best fit to the measured values using the Rashba SOC as a fit parameter, yielding λR ≈ 15 ± 5 meV. Here we are therefore able to measure both the Rashba and Ising SOC terms independently from the analysis of our data, emerging from the ability to resolve the LL structure in detail. This is the main result of this work. Using this λR value, we then compare the measured gap evolution with B for ν = 5, 6, and 7 to the single particle theory. The ν = 5 and 7 gaps’ behavior is close to the theory’s predictions, whereas the ν = 6 gap shows significant deviations. From these deviations, we determine the Coulomb interactions representing the exchange splitting for opposite spins for states derived from the K+ and K− points. We find that the
interaction energy scales vary approximately linearly with B over the measured range, consistent with previously found behavior.31,32 The ν = 5 and ν = 7 gaps differ slightly which may originate from differences in exchange energy due to the Rashba coupling. Finally, we compare the crossing points to the single particle model, using λR as a fit parameter, which yields 20 ± 1 meV, somewhat larger but in reasonable agreement to that found using the gap closing rates. This suggests that the crossing points are not too strongly affected by Coulomb interactions. Our devices were fabricated using a dry transfer and stacking method.33 Both BLG and few-layer WSe2 flakes were first exfoliated from bulk crystals and then stacked and encapsulated between atomically flat hBN layers. The interlayer BLG/ WSe2 twist angle was ∼15° as determined by the layer-edge alignment. After transfer, the stack was deposited on a Si wafer capped with 285 nm of SiO2 and vacuum annealed at 360°C for 1 h. To further promote the interlayer coupling of BLG and WSe2, we used an AFM tip to controllably squeeze out trapped contaminants from the stack.23 A clean region was identified by AFM and the rest was etched by a mixture of CHF3 (40 sccm) and O2 (4 sccm) gas into a Hall bar geometry. After the etch process, Cr/Au (5 nm/70 nm) electrodes were deposited to make one-dimensional edge contacts,33 and a metal top gate was added by an Al2O3/Au deposition step. The left inset in Figure 1a shows an optical image of the device before the top gate fabrication (located within the dotted rectangle), whereas the right inset in Figure 1a shows the layer stacking diagram. Completed devices were measured in a variable-temperature flowing gas 4He cryostat. All fourterminal resistance measurements were performed with standard lock-in methods at a base temperature of ∼1.5 K unless noted. We used the highly doped silicon substrates as a global bottom gate. In conjunction with the top gate, this device geometry allows independent control of the carrier density [n = (CBGVBG + CTGVTG)/e−n0, where CBG and CTG are the bottom and top gate capacitance per area, respectively, VBG and VTG are the bottom and top gate voltage, respectively, e is the electron charge, and n0 is residual charge due to doping] and the applied displacement field [D = (CBGVBG − 7029
DOI: 10.1021/acs.nanolett.9b02445 Nano Lett. 2019, 19, 7028−7034
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Figure 2. Transitions at zeroth Landau level. (a) Left panel: dRxy/dν versus n and D at B = 8 T. Right panel: Vertical line cuts passing LL crossing points along the dashed lines in the left panel at ν = ±3. Maxima indicate the crossing points. (b) Measured D values of LL crossing points versus B for N = 0 and 1. Open/filled circles stand for positive/negative filling factors. (c,d) Comparison between measured LL crossing points and our simulated values from the single particle model at B = 6 and 10 T, respectively. [Crossing points at B = 6 T are extracted from SI Figure S2a.].
CTGVTG)/2ε0 − D0, where D0 is the residual displacement field]. The longitudinal resistance Rxx versus VBG with VTG = 0 is shown in the Figure 1a main panel for device D1. From this data, we determine the carrier mobility to be ∼110 000 cm2 V−1 s−1 at low temperature, comparable to the high mobility achieved in typical BLG/BN systems. Under the application of a magnetic field, a color plot of Rxx versus n and B shows a Landau fan pattern [Figure 1b], consistent with the QHE in the bilayer graphene observed previously [see Supporting Information (SI) Figure S1 for more details]. The primary gaps occur at filling factors ν = ±4m, where m is an integer. This is accounted for by the bilayer LL spectrum, given by EN = ±ℏω N (N − 1) , where ω = qB/m with q as the electric charge and m as the effective mass, and N is the orbital quantum number index.34,35 These LLs are 4-fold degenerate due to spin and valley degeneracies but are split by perturbations such as interactions and Zeeman splitting.36,37 Analyzing the QHE data, we extract the values of gate capacitances CBG = 13.5 nF/cm2 and CTG = 72.5 nF/cm2. For B larger than 6 T, we observe all integer QHE states, indicating the high quality of our sample. To investigate the layer-selective spin−orbit proximity coupling, we studied the effects of varying D. The Figure 1c lower panel shows a color plot of Rxx versus n and D, taken at B = 10 T. Similar data at B = 6 and 8 T are shown in SI Figure S2. (An artifact likely due to contact resistance and possible formation of resistive p−n junctions causes a downwardsloping feature in the plot obscuring the QHE signatures, especially at high B. An example of one of these features is marked by a white arrow. Here, we focus on the data outside this region.) The upper panel in Figure 1c shows Rxx plotted
versus n for D = 0, yielding a series of Rxx minima due to the LLs in the BLG. All the odd filling states are present at D = 0 in contrast to BLG without WSe2.27,28 As D varies at fixed n, some of the Rxx minima vanish and then reemerge, indicating the closing and reopening of inter-LL gaps at LL crossing points. In Figure 1d, line traces of Rxx minima at ν = 5 and 6 fillings show peaks at particular D values, enabling extraction of the crossing points. Although the ν = 7 curve does not go through a maximum in the selected range, the upward slope suggests that a crossing point exists near D = −0.15 V/nm. As we find that Rxy is impacted less than Rxx by the contact artifact (see Figure S3), to ensure accuracy, we also calculate the derivative of Rxy with respect to n. This produces similar peaks as D varies, enabling the extraction of the same D values for the LL crossings and enables determining crossing points where Rxx data is unavailable due to the contact artifact. The N = 0, 1 LLs in an isolated bilayer are degenerate in the single particle picture except for Zeeman splitting but interactions lift all degeneracies.24,29,36,38−41 Adding SOC is expected to modify both the inter-LL gaps and crossing points. We zoom in to the dRxy/dν versus D and n data at B = 8 T to extract the asymmetric crossing points for the LLs. Left panel in Figure 2a shows this color plot in which the crossing points are readily visible, such as that marked by the white arrow. From line traces along fixed n at particular filling factors (for example, ν = ±3 shown in the right panel), we find and plot the D values at the crossings in Figure 2b. For most filling factors, the crossing points have the same D magnitude but with their sign depending on the sign of ν. The exception is ν = ±2 in which the ν = 2 state crossing occurs at a smaller D value than ν = −2. To take initial steps toward understanding this behavior, following ref 20, we assume a phenomenological single-particle 7030
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Figure 3. LL energies and broken symmetry gaps in a BLG/WSe2 system. (a) Thermal activation plots at various B values for ν = 6 states at D = 0. (b) Measured thermal activation gaps Δ − Γ versus B at D = 0 for ν = 5, 6, and 7, where Γ is the LL width (left axis). Dotted lines are linear fits of gap sizes versus B. The black dotted line stands for simulated gap size for ν = 5 without broadening (right axis). Inset: single particle model for all three gaps ν = 5, 6, and 7. (c) Measured gaps Δ − Γ versus D at B = 10 T for ν = 6. The missing data points are at the LL crossing point. By fitting the slopes of the gap changes with D (dashed lines), we extract gap closing rates near LL crossing points and compare them with simulated values, as shown in Table 1. (Gap closing rates for other filling factors in addition to ν = 6 are extracted from data shown in SI Figure S7.)
for ν = ±5 and ±7.26−28 Moreover, under inversion symmetry the crossing points are symmetric in D, which is not the case in the SOC device. Thus, the asymmetry in D and fully broken degeneracies are clear evidence of the inversion symmetry breaking caused by the presence of the WSe2. We now turn to quantitatively estimating λ and λR from the data. The crossing points for ν = ±3 are expected to be unaffected by Coulomb interactions22,29 because although Coulomb interactions can change the order of filling the Landau levels within the zero energy octet, they do not affect which Landau level fills first or last.22,29 They are also insensitive to the Rashba coupling.20 The crossing points of ν = ±3 are relatively insensitive to B [Figure 2b]. To understand this, we note that in a magnetic field the nearly degenerate zero-energy LL octet states are strongly layer and valley polarized with layer and valley degrees of freedom coupled. The lower and upper layer states are concentrated in the K+ and K− valleys, respectively. Because the WSe2 contacts the bottom layer, yielding layer selective SOC, SOC only strongly affects the K+ valley states. For these states, a positive λ acts against the Zeeman splitting, whereas a negative λ enhances it. The lowest energy states there are spin up when λ < EZ and spin down when λ > EZ. In contrast, the upper layer K− valley states’ spin splitting is essentially determined only by the Zeeman effect with the lowest energy states spin up.
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model of SOC incorporating both an Ising term, HI = 2 ξλsz 1
and a Rashba term, HR = 2 λR (ξσxsy‐σysz), where λ is an energy scale describing the out-of-plane spin splitting caused by SOC, λR is an energy scale describing the in-plane Rashba SOC, the σx,y,z are Pauli matrices for the A and B sublattices, the sx,y,z are Pauli matrices for spin, and ξ = +1 for states near the K+ point and −1 for the states near the K− point. The SOC term breaks inversion symmetry. On the other hand, the Hamiltonian is symmetric under time reversal as this not only changes the sign of η but the signs of the spins while not exchanging the sublattices A and B. All bilayer single particle parameters such as nearest neighbor and interlayer hopping are taken from ref 1 20. We also add a Zeeman term HZ = − 2 EZ sz ,20 where EZ = gμBB is the Zeeman energy with g ≈ 2 the electron g-factor and μB is the Bohr magneton, and we account for the interlayer potential u caused by D (LL energies versus u for B = 8 T are plotted in Figure S4). We solve for the eigenvalues and eigenstates numerically as a function of the parameters B, λR, λ, and u (see SI for additional details). When D = 0, all the degeneracies are lifted as a result of SOC coupling in both the model and in the data as shown in the upper panel of Figure 1c. In contrast, although interactions potentially lift degeneracies, in non-SOC BLG, K+−K− degeneracy yields two doubly degenerate LLs in each quartet so that the gaps vanish at D = 0 7031
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Figure 4. Corrections to the single particle model due to Coulomb interactions. (a) Schematic representation of the broken symmetry gaps for N = 2 at D = 0. ΔK and ΔK′ stands for the proximity induced Ising and Rashba SOC, causing different splitting in the K+ and K− valleys. JK+ and JK− are the Coulomb interactions for the electrons with opposite spins in the same valley. (b) Extracted differential ΔJK− relative to its value at B = 6 T at D = 0.
λR = 22 meV. However, these crossing points are in principle still potentially affected by Coulomb interactions. To elucidate the role of Coulomb interactions in the behavior of the system, we determine the transport gaps by measuring the temperature dependence of Rxx in a number of valleys versus D for device D1. An example of this behavior is shown in Figure 3a which shows Arrhenius plots for ν = 6 at D = 0 versus B. We then fit the Rxx minima against 1/T with the ÅÄÅ Δ − Γ ÑÉÑ equation R = R 0 expÅÅÅ− 2k T ÑÑÑ to extract the quantity Δ − Γ, ÅÇ B Ñ Ö where Δ is the gap and Γ (full width at half-maximum) is the LL width. Figure 3b shows the extracted Δ − Γ versus B for ν = 5, 6, and 7. To make a comparison to the single particle model above, we consider the rate of change of Δ − Γ with D [i.e., d(Δ − Γ)/dD = dΔ/dD assuming constant Γ] near a gap closing, such as shown in Figure 3c. This is insensitive to Γ, as well as Coulomb interactions within a Hartree−Fock picture. Table 1 in Figure 3 shows the measured values for a number of crossings. Values on both sides of the crossings are averaged if both are obtained. We then compute the expected dΔ/dD at each crossing and using only λR as a free parameter minimize the sum of the squared differences between all the expected and measured crossing rates with respect to λR. We find a best fit value of λR ≈ 15 ± 5 meV for the data shown. The computed values using this λR value are tabulated in Figure 3 Table 1 column 4 which show reasonable agreement to the measured values. However, deviations exist which may, for example, stem from measurement uncertainty in the gap closing rates. This indicates that the λR obtained from this method should be considered only an estimate and characteristically at ∼15 meV, consistent with the range of previously found values, for example, refs 9 and 15. Using this extracted value of λR, we compute the expected behavior of the ν = 5 gap and plot it as the dashed line in Figure 3b without free parameters. Fitting a straight line to the ν = 5 gap data in Figure 3b yields a slope of 0.11 ± 0.02 meV/ T, very close to the theoretical slope of ∼gμB = 0.115 meV/T, indicating that Coulomb interactions have only a small effect on the gap. The ν = 7 gap is slightly larger but similar in magnitude. On the other hand, the fitted slope of 0.245 meV/ T of the ν = 6 gap is significantly larger than gμB and also
For λ > EZ, the ν = ±3 crossing states therefore have the same orbital but opposite spins (see SI Figure S4). The crossing points are B-dependent as found in ref 22 (see also Figure S5). In this case, the ν = ±3 crossing points both occur at D = 0 when λ = 2EZ and then switch their signs of D for larger B.22 In contrast, for λ < EZ (including negative λ values) the ν = ±3 crossing states have the same orbital and spin (see SI Figure S4). The crossing points are thus B-independent. (We note that in the zero-energy LLs Rashba coupling is expected to contribute negligibly small level shifts ∼λR3/γ12, where γ1 = 0.361 eV.20 An exception to this can occur at a potential level crossing within the same valley in which case the Rashba term can produce an anticrossing at a noticeably larger energy scale. However, this does not affect the ν = ±3 crossings.) The observed B-insensitivity of the ν = ±3 crossings thus implies that λ < EZ in device D1. By carefully comparing the ν = ±3 crossing D values with the theory, we find λ = −2.2 ± 0.1 meV. Note that at the maximum B measured, B = 10 T, EZ ≈ 1.1 meV, thus if λ were positive but of the same magnitude we would have instead λ > EZ. This would lead to a strong B dependence of the ν = ±3 crossing points reaching D = 0 when EZ = λ/2 = 1.1 meV corresponding to B ≈ 10 T, contradicting our observations. A negative λ is also indicated by the crossings for ν = 3 and 5 having the same sign of D (see SI Figure S4) at B = 8 T. This is consistent with theoretical predictions that negative λ occurs for certain ranges of graphene−WSe2 twist angles.30 (Similar results were also obtained from another device D2, shown in SI Figure S6.) For the Rashba coupling, expected to be positive regardless of twist angle,30 we measure the crossing points’ D-values for ν = ±5, ±6, and ±7. Figure 2c plots this data for B = 6 T and Figure 2d plots the corresponding data for B = 10 T. Using λ = −2.2 meV as obtained above and using λR as the only free parameter, we minimize the sum of all the squared deviations between the predicted single particle values and the measured data, obtaining λR ∼ 20 ± 1 meV. Figure 2c,d shows the resulting model values using the best fit λR plotted as open triangles. The single particle model with this value of λR yields a reasonable agreement to the measured values. Device D2 (SI Figure S6) also showed similar results with λ = −2.6 meV and 7032
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increases with B. In a single particle picture, the quartet of LLs associated with these gaps consists of states that can be labeled by a K+ or K− valley index and spin. The K+ and K− states are split by the SOC into nearly up and down spin states (with slight canting), which are oppositely directed in the two valleys, effectively. The single particle calculation shows that although the Rashba SOC reduces the total splitting, the ν = 5 gap varies linearly with B with a slope ≈ gμΒ, as observed. The measured energy is lower than the predicted single particle splitting, which we attribute to an LL width of ∼0.4 meV. The results of the single particle model for all three gaps ν = 5, 6, and 7 are plotted over a wider range of B in the Figure 3b inset. Although this picture produces reasonable results over the measured B range for Δ5 and Δ7, it predicts that Δ6 starts at a splitting set by the SOC and decreases with B, contrary to what is observed. We therefore consider a model that includes Coulomb interactions. SOC breaks the K+-K− symmetry, leading to potentially different exchange energies in the two valleys. Additionally, there could be an exchange coupling between the K+ and K− states. However, based on the close agreement in the behavior of the Δ5 and Δ7 gaps to the single particle picture we assume this is negligible. Thus, we model the Coulomb interaction with two parameters, which are JK+ and JK−, corresponding to the exchange energy to flip a spin within an LL in the K+ or K− valleys with an effective single particle picture shown schematically in Figure 4a. The similar energy scales measured for Δ5 and Δ7 indicate that JK+ ≈ JK−. We obtain JK− by taking the total gap Δ6 to be Δ6−sp + JK−, where Δ6−sp is the expected single particle gap at D = 0, plotted in the inset to Figure 4b, and rearranging it to find JK− = Δ6 − Δ6−sp. Because of the unknown Coulomb interactions outside of the measured range and the uncertainty in the LL width, this quantity is plotted in Figure 4b as a differential ΔJK− relative to its value at B = 6 T. The behavior is approximately linear in B. Although it may be initially expected that JK− is proportional to the Coulomb energy scale e2/εlB, where lB is the magnetic length and ε is the dielectric constant, this result is in agreement with previous experiments that also find an approximately linear in B Coulomb interaction.31,32 This may arise from changes to the dielectric constant with B.42 However, more work will be required to understand the origin of this behavior. In conclusion, we have measured high mobility bilayer graphene samples in which the spin−orbit interaction is induced by proximity coupling to WSe2. The Ising parameter is measured to be λ ≈ −2.2 meV, whereas the Rashba parameter is ∼15 meV. The SOC breaks all degeneracies of the LLs at D = 0, and although the crossing points versus D are qualitatively in agreement with a single particle model both the gaps and the crossing points can be affected by Coulomb interactions. We measure the Coulomb energy corrections for ν = 6 and find it is approximately linear in B over the measured range, in accordance with previous results on non-SOC bilayer graphene. The quantum Hall effect is a useful probe of SOC in bilayer graphene. In future work, this approach may be used to test theories in which both the Ising and Rashba SOC are tunable by the twist angle between the graphene and WSe2,30,43 potentially affording a unique approach to controlling SOC in materials.
Letter
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.9b02445.
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Additional discussion of single particle model and data analysis as well as additional figures (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Phone: 614-292-0375. ORCID
Dongying Wang: 0000-0002-9674-3532 Chun Ning Lau: 0000-0003-2159-6723 Marc Bockrath: 0000-0002-7000-1442 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Jyoti Katoch for helpful discussions. This research funded by DOE ER 46940-DE-SC0010597. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan, A3 Foresight by JSPS, and the CREST(JPMJCR15F3), JST.
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REFERENCES
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Nano Letters T.; Eda, G.; Castro Neto, A. H.; Ö zyilmaz, B. Spin−orbit proximity effect in graphene. Nat. Commun. 2014, 5, 4875. (13) Yang, B.; Tu, M.-F.; Kim, J.; Wu, Y.; Wang, H.; Alicea, J.; Wu, R.; Bockrath, M.; Shi, J. Tunable spin−orbit coupling and symmetryprotected edge states in graphene/WS2. 2D Mater. 2016, 3 (3), No. 031012. (14) Yang, B.; Lohmann, M.; Barroso, D.; Liao, I.; Lin, Z.; Liu, Y.; Bartels, L.; Watanabe, K.; Taniguchi, T.; Shi, J. Strong electron-hole symmetric Rashba spin-orbit coupling in graphene/monolayer transition metal dichalcogenide heterostructures. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 96 (4), No. 041409. (15) Wakamura, T.; Reale, F.; Palczynski, P.; Guéron, S.; Mattevi, C.; Bouchiat, H. Strong Anisotropic Spin-Orbit Interaction Induced in Graphene by Monolayer WS2. Phys. Rev. Lett. 2018, 120 (10), 106802. (16) Zihlmann, S.; Cummings, A. W.; Garcia, J. H.; Kedves, M.; Watanabe, K.; Taniguchi, T.; Schönenberger, C.; Makk, P. Large spin relaxation anisotropy and valley-Zeeman spin-orbit coupling in WSe2/ graphene/h-BN heterostructures. Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 97 (7), No. 075434. (17) Asmar, M. M.; Ulloa, S. E. Symmetry-breaking effects on spin and electronic transport in graphene. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 91 (16), 165407. (18) Gmitra, M.; Fabian, J. Graphene on transition-metal dichalcogenides: A platform for proximity spin-orbit physics and optospintronics. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 92 (15), 155403. (19) Gmitra, M.; Fabian, J. Proximity Effects in Bilayer Graphene on Monolayer WSe2: Field-Effect Spin Valley Locking, Spin-Orbit Valve, and Spin Transistor. Phys. Rev. Lett. 2017, 119 (14), 146401. (20) Khoo, J. Y.; Levitov, L. Tunable quantum Hall edge conduction in bilayer graphene through spin-orbit interaction. Phys. Rev. B: Condens. Matter Mater. Phys. 2018, 98 (11), 115307. (21) Khoo, J. Y.; Morpurgo, A. F.; Levitov, L. On-Demand Spin− Orbit Interaction from Which-Layer Tunability in Bilayer Graphene. Nano Lett. 2017, 17 (11), 7003−7008. (22) Island, J.; Cui, X.; Lewandowski, C.; Khoo, J.; Spanton, E.; Zhou, H.; Rhodes, D.; Hone, J.; Taniguchi, T.; Watanabe, K.; Young, A. F.; et al. Spin-orbit driven band inversion in bilayer graphene by van der Waals proximity effect. Nature 2019, 571, 85−89. (23) Rosenberger, M. R.; Chuang, H.-J.; McCreary, K. M.; Hanbicki, A. T.; Sivaram, S. V.; Jonker, B. T. Nano-“Squeegee” for the Creation of Clean 2D Material Interfaces. ACS Appl. Mater. Interfaces 2018, 10 (12), 10379−10387. (24) Weitz, R. T.; Allen, M. T.; Feldman, B. E.; Martin, J.; Yacoby, A. Broken-Symmetry States in Doubly Gated Suspended Bilayer Graphene. Science 2010, 330 (6005), 812. (25) Velasco, J., Jr; Jing, L.; Bao, W.; Lee, Y.; Kratz, P.; Aji, V.; Bockrath, M.; Lau, C. N.; Varma, C.; Stillwell, R.; Smirnov, D.; Zhang, F.; Jung, J.; MacDonald, A. H. Transport spectroscopy of symmetrybroken insulating states in bilayer graphene. Nat. Nanotechnol. 2012, 7, 156. (26) Maher, P.; Wang, L.; Gao, Y.; Forsythe, C.; Taniguchi, T.; Watanabe, K.; Abanin, D.; Papić, Z.; Cadden-Zimansky, P.; Hone, J.; Kim, P.; Dean, C. R. Tunable fractional quantum Hall phases in bilayer graphene. Science 2014, 345 (6192), 61. (27) Lee, K.; Fallahazad, B.; Xue, J.; Dillen, D. C.; Kim, K.; Taniguchi, T.; Watanabe, K.; Tutuc, E. Chemical potential and quantum Hall ferromagnetism in bilayer graphene. Science 2014, 345 (6192), 58. (28) Pan, C.; Wu, Y.; Cheng, B.; Che, S.; Taniguchi, T.; Watanabe, K.; Lau, C. N.; Bockrath, M. Layer Polarizability and Easy-Axis Quantum Hall Ferromagnetism in Bilayer Graphene. Nano Lett. 2017, 17 (6), 3416−3420. (29) Hunt, B. M.; Li, J. I. A.; Zibrov, A. A.; Wang, L.; Taniguchi, T.; Watanabe, K.; Hone, J.; Dean, C. R.; Zaletel, M.; Ashoori, R. C.; Young, A. F. Direct measurement of discrete valley and orbital quantum numbers in bilayer graphene. Nat. Commun. 2017, 8 (1), 948.
(30) David, A.; Rakyta, P.; Kormányos, A.; Burkard, G., Induced spin-orbit coupling in twisted graphene-TMDC heterobilayers: twistronics meets spintronics. 2019, arXiv preprint arXiv:1905.08688. (31) Martin, J.; Akerman, N.; Ulbricht, G.; Lohmann, T.; Smet, J. H.; von Klitzing, K.; Yacoby, A. Observation of electron−hole puddles in graphene using a scanning single-electron transistor. Nat. Phys. 2008, 4, 144. (32) Shi, Y.; Lee, Y.; Che, S.; Pi, Z.; Espiritu, T.; Stepanov, P.; Smirnov, D.; Lau, C. N.; Zhang, F. Energy Gaps and Layer Polarization of Integer and Fractional Quantum Hall States in Bilayer Graphene. Phys. Rev. Lett. 2016, 116 (5), No. 056601. (33) Wang, L.; Meric, I.; Huang, P. Y.; Gao, Q.; Gao, Y.; Tran, H.; Taniguchi, T.; Watanabe, K.; Campos, L. M.; Muller, D. A.; Guo, J.; Kim, P.; Hone, J.; Shepard, K. L.; Dean, C. R. One-Dimensional Electrical Contact to a Two-Dimensional Material. Science 2013, 342 (6158), 614. (34) McCann, E.; Fal’ko, V. I. Landau-Level Degeneracy and Quantum Hall Effect in a Graphite Bilayer. Phys. Rev. Lett. 2006, 96 (8), No. 086805. (35) Novoselov, K. S.; McCann, E.; Morozov, S. V.; Fal’ko, V. I.; Katsnelson, M. I.; Zeitler, U.; Jiang, D.; Schedin, F.; Geim, A. K. Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene. Nat. Phys. 2006, 2, 177. (36) Zhao, Y.; Cadden-Zimansky, P.; Jiang, Z.; Kim, P. Symmetry Breaking in the Zero-Energy Landau Level in Bilayer Graphene. Phys. Rev. Lett. 2010, 104 (6), No. 066801. (37) Dean, C. R.; Young, A. F.; Meric, I.; Lee, C.; Wang, L.; Sorgenfrei, S.; Watanabe, K.; Taniguchi, T.; Kim, P.; Shepard, K. L.; Hone, J. Boron nitride substrates for high-quality graphene electronics. Nat. Nanotechnol. 2010, 5, 722. (38) Young, A. F.; Dean, C. R.; Wang, L.; Ren, H.; CaddenZimansky, P.; Watanabe, K.; Taniguchi, T.; Hone, J.; Shepard, K. L.; Kim, P. Spin and valley quantum Hall ferromagnetism in graphene. Nat. Phys. 2012, 8, 550. (39) Alicea, J.; Fisher, M. P. A. Graphene integer quantum Hall effect in the ferromagnetic and paramagnetic regimes. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74 (7), No. 075422. (40) Lambert, J.; Côté, R. Quantum Hall ferromagnetic phases in the Landau level N = 0 of a graphene bilayer. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 87 (11), 115415. (41) Li, J.; Tupikov, Y.; Watanabe, K.; Taniguchi, T.; Zhu, J. Effective Landau Level Diagram of Bilayer Graphene. Phys. Rev. Lett. 2018, 120 (4), No. 047701. (42) Shizuya, K. Structure and the Lamb-shift-like quantum splitting of the pseudo-zero-mode Landau levels in bilayer graphene. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86 (4), No. 045431. (43) Li, Y.; Koshino, M. Twist-angle dependence of the proximity spin-orbit coupling in graphene on transition-metal dichalcogenides. Phys. Rev. B: Condens. Matter Mater. Phys. 2019, 99 (7), No. 075438.
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Cite This: Nano Lett. 2019, 19, 7028−7034
Quantum Hall Effect Measurement of Spin−Orbit Coupling Strengths in Ultraclean Bilayer Graphene/WSe2 Heterostructures Dongying Wang,† Shi Che,† Guixin Cao,† Rui Lyu,‡ Kenji Watanabe,§ Takashi Taniguchi,§ Chun Ning Lau,† and Marc Bockrath*,† †
Department of Physics, The Ohio State University, Columbus, Ohio 43210, United States Department of Physics and Astronomy, University of California, Riverside, California 92521, United States § National Institute for Materials Science, Namiki Tsukuba Ibaraki 305-0044 Japan Downloaded via UNIV OF NEW ENGLAND on October 22, 2019 at 20:27:37 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
‡
S Supporting Information *
ABSTRACT: We study proximity-induced spin−orbit coupling (SOC) in bilayer graphene/few-layer WSe2 heterostructure devices. Contact mode atomic force microscopy (AFM) cleaning yields ultraclean interfaces and high-mobility devices. In a perpendicular magnetic field, we measure the quantum Hall effect to determine the Landau level structure in the presence of out-of-plane Ising and in-plane Rashba SOC. A distinct Landau level crossing pattern emerges when tuning the charge density and displacement field independently with dual gates, originating from a layer-selective SOC proximity effect. Analyzing the Landau level crossings and measured inter-Landau level energy gaps yields the proximity-induced SOC energy scale. The Ising SOC is ∼2.2 meV, 100 times higher than the intrinsic SOC in graphene, whereas its sign is consistent with theories predicting a dependence of SOC on interlayer twist angle. The Rashba SOC is ∼15 meV. Finally, we infer the magnetic field dependence of the inter-Landau level Coulomb interactions. These ultraclean bilayer graphene/WSe2 heterostructures provide a high mobility system with the potential to realize novel topological electronic states and manipulate spins in nanostructures. KEYWORDS: Graphene, transition-metal dichalcogenide, spin−orbit coupling, quantum transport fforts to control spin and electronic ground state topology in devices has driven an intensive study of spin−orbit coupling in materials.1,2 Graphene is a two-dimensional (2D) material with excellent electronic properties but small spin− orbit coupling (SOC),3−6 motivating efforts to increase it.7−21 For example, adatoms such as hydrogen or transition metals increase graphene’s SOC,7,8 however these add disorder and lower charge mobility.7 In another approach, coupling graphene to 2D materials with strong SOC such as transition metal dichalcogenides also increases graphene’s SOC,9−16,18,19,21 whereas the interfacing of two crystalline systems adds less intrinsic disorder. Such studies have shown weak antilocalization signatures of spin−orbit coupling9,10,13−16 as well as spin Hall signatures.12 Shubnikov-de Haas oscillations were used to infer a dominant Rashba SOC of magnitude ∼10−15 meV.9 However, electrical transport studies in high mobility devices on the quantum Hall effect are limited which can act as a precise simultaneous probe of both
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© 2019 American Chemical Society
Ising and Rashba SOC in graphene and potentially realize new topological ground states.19,20,22 Moreover, although recent progress has been made by a combination of capacitance and some transport measurements,22 inter-Landau level energy gaps have not been measured, nor the Rashba SOC or magnetic field dependence of the inter-Landau level Coulomb interactions. Here we report coupling bilayer graphene (BLG) via layer stacking to WSe2, a 2D semiconductor with strong SOC, in hexagonal BN (hBN) encapsulated devices. We obtain high mobility devices (up to ∼110 000 cm2 V−1 s−1) by squeezing contaminants away from device areas using contact AFM.23 Our devices are dual gated, allowing independent tuning of the Received: June 15, 2019 Revised: September 15, 2019 Published: September 17, 2019 7028
DOI: 10.1021/acs.nanolett.9b02445 Nano Lett. 2019, 19, 7028−7034
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Figure 1. Device structure and SOC modified LL properties. (a) Main panel: Rxx versus VBG at T = 1.5 K with VTG = 0. Left inset: Optical image of a BLG/WSe2 device. The dashed outline indicates the top gated region before metalization. Scale bar is 2 μm. Right inset: Schematic diagram of the layer stack. (b) Rxx versus VBG and magnetic field B showing a Landau Fan pattern. Quantum Hall states at all integer filling factors of the Landau levels are clearly visible due to the full degeneracy lifting. (c) Top panel: Rxx versus n at D = 0. Bottom panel: Color plot of Rxx versus n and D at B = 10 T. (d) Vertical line cuts passing Landau level crossing points along the dashed lines in panel c. Each curve represents a fixed filling factor for ν = 5, 6, and 7.
charge density n and perpendicular displacement field D. At low temperatures, under a perpendicular magnetic field B, we observe the quantum Hall effect (QHE) with all degeneracies lifted for D = 0, resulting from inversion symmetry breaking by the WSe2 layer contacting the graphene. Applying a nonzero D yields a finite interlayer potential difference, enabling us to observe Landau level (LL) crossings24−28 that differ in D values from isolated BLG. We obtained high quality data from two devices, D1 and D2. To understand this behavior, we compare our results to a single-particle theory that includes an out-of-plane Ising SOC λ as well as an in-plane Rashba SOC λR.20 For the nearly degenerate zero energy LL octet, most crossings deviate from the single-particle picture due to Coulomb interactions except for the ν = ±3 crossings.20,22,29 From these we extract λ ∼ −2.2 ± 0.1 meV, where the sign indicates the direction of the effective out-of-plane magnetic field from the SOC in the bilayer graphene at the K+ point (see SI for additional details) of the Brillouin zone (with the opposite direction at the K− point),20 consistent with recent theoretical work predicting that λ can be negative depending on the graphene/WSe2 twist angle.30 To gain further insight into the device behavior, we measure the inter-LL gaps using temperature-dependent transport measurements. These gaps are potentially affected by both the LL width as well as Coulomb interactions. We therefore study the rate of gap closure with the displacement field which is independent of these within a Hartree−Fock model. Measuring these rates for several gaps, we find the best fit to the measured values using the Rashba SOC as a fit parameter, yielding λR ≈ 15 ± 5 meV. Here we are therefore able to measure both the Rashba and Ising SOC terms independently from the analysis of our data, emerging from the ability to resolve the LL structure in detail. This is the main result of this work. Using this λR value, we then compare the measured gap evolution with B for ν = 5, 6, and 7 to the single particle theory. The ν = 5 and 7 gaps’ behavior is close to the theory’s predictions, whereas the ν = 6 gap shows significant deviations. From these deviations, we determine the Coulomb interactions representing the exchange splitting for opposite spins for states derived from the K+ and K− points. We find that the
interaction energy scales vary approximately linearly with B over the measured range, consistent with previously found behavior.31,32 The ν = 5 and ν = 7 gaps differ slightly which may originate from differences in exchange energy due to the Rashba coupling. Finally, we compare the crossing points to the single particle model, using λR as a fit parameter, which yields 20 ± 1 meV, somewhat larger but in reasonable agreement to that found using the gap closing rates. This suggests that the crossing points are not too strongly affected by Coulomb interactions. Our devices were fabricated using a dry transfer and stacking method.33 Both BLG and few-layer WSe2 flakes were first exfoliated from bulk crystals and then stacked and encapsulated between atomically flat hBN layers. The interlayer BLG/ WSe2 twist angle was ∼15° as determined by the layer-edge alignment. After transfer, the stack was deposited on a Si wafer capped with 285 nm of SiO2 and vacuum annealed at 360°C for 1 h. To further promote the interlayer coupling of BLG and WSe2, we used an AFM tip to controllably squeeze out trapped contaminants from the stack.23 A clean region was identified by AFM and the rest was etched by a mixture of CHF3 (40 sccm) and O2 (4 sccm) gas into a Hall bar geometry. After the etch process, Cr/Au (5 nm/70 nm) electrodes were deposited to make one-dimensional edge contacts,33 and a metal top gate was added by an Al2O3/Au deposition step. The left inset in Figure 1a shows an optical image of the device before the top gate fabrication (located within the dotted rectangle), whereas the right inset in Figure 1a shows the layer stacking diagram. Completed devices were measured in a variable-temperature flowing gas 4He cryostat. All fourterminal resistance measurements were performed with standard lock-in methods at a base temperature of ∼1.5 K unless noted. We used the highly doped silicon substrates as a global bottom gate. In conjunction with the top gate, this device geometry allows independent control of the carrier density [n = (CBGVBG + CTGVTG)/e−n0, where CBG and CTG are the bottom and top gate capacitance per area, respectively, VBG and VTG are the bottom and top gate voltage, respectively, e is the electron charge, and n0 is residual charge due to doping] and the applied displacement field [D = (CBGVBG − 7029
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Figure 2. Transitions at zeroth Landau level. (a) Left panel: dRxy/dν versus n and D at B = 8 T. Right panel: Vertical line cuts passing LL crossing points along the dashed lines in the left panel at ν = ±3. Maxima indicate the crossing points. (b) Measured D values of LL crossing points versus B for N = 0 and 1. Open/filled circles stand for positive/negative filling factors. (c,d) Comparison between measured LL crossing points and our simulated values from the single particle model at B = 6 and 10 T, respectively. [Crossing points at B = 6 T are extracted from SI Figure S2a.].
CTGVTG)/2ε0 − D0, where D0 is the residual displacement field]. The longitudinal resistance Rxx versus VBG with VTG = 0 is shown in the Figure 1a main panel for device D1. From this data, we determine the carrier mobility to be ∼110 000 cm2 V−1 s−1 at low temperature, comparable to the high mobility achieved in typical BLG/BN systems. Under the application of a magnetic field, a color plot of Rxx versus n and B shows a Landau fan pattern [Figure 1b], consistent with the QHE in the bilayer graphene observed previously [see Supporting Information (SI) Figure S1 for more details]. The primary gaps occur at filling factors ν = ±4m, where m is an integer. This is accounted for by the bilayer LL spectrum, given by EN = ±ℏω N (N − 1) , where ω = qB/m with q as the electric charge and m as the effective mass, and N is the orbital quantum number index.34,35 These LLs are 4-fold degenerate due to spin and valley degeneracies but are split by perturbations such as interactions and Zeeman splitting.36,37 Analyzing the QHE data, we extract the values of gate capacitances CBG = 13.5 nF/cm2 and CTG = 72.5 nF/cm2. For B larger than 6 T, we observe all integer QHE states, indicating the high quality of our sample. To investigate the layer-selective spin−orbit proximity coupling, we studied the effects of varying D. The Figure 1c lower panel shows a color plot of Rxx versus n and D, taken at B = 10 T. Similar data at B = 6 and 8 T are shown in SI Figure S2. (An artifact likely due to contact resistance and possible formation of resistive p−n junctions causes a downwardsloping feature in the plot obscuring the QHE signatures, especially at high B. An example of one of these features is marked by a white arrow. Here, we focus on the data outside this region.) The upper panel in Figure 1c shows Rxx plotted
versus n for D = 0, yielding a series of Rxx minima due to the LLs in the BLG. All the odd filling states are present at D = 0 in contrast to BLG without WSe2.27,28 As D varies at fixed n, some of the Rxx minima vanish and then reemerge, indicating the closing and reopening of inter-LL gaps at LL crossing points. In Figure 1d, line traces of Rxx minima at ν = 5 and 6 fillings show peaks at particular D values, enabling extraction of the crossing points. Although the ν = 7 curve does not go through a maximum in the selected range, the upward slope suggests that a crossing point exists near D = −0.15 V/nm. As we find that Rxy is impacted less than Rxx by the contact artifact (see Figure S3), to ensure accuracy, we also calculate the derivative of Rxy with respect to n. This produces similar peaks as D varies, enabling the extraction of the same D values for the LL crossings and enables determining crossing points where Rxx data is unavailable due to the contact artifact. The N = 0, 1 LLs in an isolated bilayer are degenerate in the single particle picture except for Zeeman splitting but interactions lift all degeneracies.24,29,36,38−41 Adding SOC is expected to modify both the inter-LL gaps and crossing points. We zoom in to the dRxy/dν versus D and n data at B = 8 T to extract the asymmetric crossing points for the LLs. Left panel in Figure 2a shows this color plot in which the crossing points are readily visible, such as that marked by the white arrow. From line traces along fixed n at particular filling factors (for example, ν = ±3 shown in the right panel), we find and plot the D values at the crossings in Figure 2b. For most filling factors, the crossing points have the same D magnitude but with their sign depending on the sign of ν. The exception is ν = ±2 in which the ν = 2 state crossing occurs at a smaller D value than ν = −2. To take initial steps toward understanding this behavior, following ref 20, we assume a phenomenological single-particle 7030
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Figure 3. LL energies and broken symmetry gaps in a BLG/WSe2 system. (a) Thermal activation plots at various B values for ν = 6 states at D = 0. (b) Measured thermal activation gaps Δ − Γ versus B at D = 0 for ν = 5, 6, and 7, where Γ is the LL width (left axis). Dotted lines are linear fits of gap sizes versus B. The black dotted line stands for simulated gap size for ν = 5 without broadening (right axis). Inset: single particle model for all three gaps ν = 5, 6, and 7. (c) Measured gaps Δ − Γ versus D at B = 10 T for ν = 6. The missing data points are at the LL crossing point. By fitting the slopes of the gap changes with D (dashed lines), we extract gap closing rates near LL crossing points and compare them with simulated values, as shown in Table 1. (Gap closing rates for other filling factors in addition to ν = 6 are extracted from data shown in SI Figure S7.)
for ν = ±5 and ±7.26−28 Moreover, under inversion symmetry the crossing points are symmetric in D, which is not the case in the SOC device. Thus, the asymmetry in D and fully broken degeneracies are clear evidence of the inversion symmetry breaking caused by the presence of the WSe2. We now turn to quantitatively estimating λ and λR from the data. The crossing points for ν = ±3 are expected to be unaffected by Coulomb interactions22,29 because although Coulomb interactions can change the order of filling the Landau levels within the zero energy octet, they do not affect which Landau level fills first or last.22,29 They are also insensitive to the Rashba coupling.20 The crossing points of ν = ±3 are relatively insensitive to B [Figure 2b]. To understand this, we note that in a magnetic field the nearly degenerate zero-energy LL octet states are strongly layer and valley polarized with layer and valley degrees of freedom coupled. The lower and upper layer states are concentrated in the K+ and K− valleys, respectively. Because the WSe2 contacts the bottom layer, yielding layer selective SOC, SOC only strongly affects the K+ valley states. For these states, a positive λ acts against the Zeeman splitting, whereas a negative λ enhances it. The lowest energy states there are spin up when λ < EZ and spin down when λ > EZ. In contrast, the upper layer K− valley states’ spin splitting is essentially determined only by the Zeeman effect with the lowest energy states spin up.
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model of SOC incorporating both an Ising term, HI = 2 ξλsz 1
and a Rashba term, HR = 2 λR (ξσxsy‐σysz), where λ is an energy scale describing the out-of-plane spin splitting caused by SOC, λR is an energy scale describing the in-plane Rashba SOC, the σx,y,z are Pauli matrices for the A and B sublattices, the sx,y,z are Pauli matrices for spin, and ξ = +1 for states near the K+ point and −1 for the states near the K− point. The SOC term breaks inversion symmetry. On the other hand, the Hamiltonian is symmetric under time reversal as this not only changes the sign of η but the signs of the spins while not exchanging the sublattices A and B. All bilayer single particle parameters such as nearest neighbor and interlayer hopping are taken from ref 1 20. We also add a Zeeman term HZ = − 2 EZ sz ,20 where EZ = gμBB is the Zeeman energy with g ≈ 2 the electron g-factor and μB is the Bohr magneton, and we account for the interlayer potential u caused by D (LL energies versus u for B = 8 T are plotted in Figure S4). We solve for the eigenvalues and eigenstates numerically as a function of the parameters B, λR, λ, and u (see SI for additional details). When D = 0, all the degeneracies are lifted as a result of SOC coupling in both the model and in the data as shown in the upper panel of Figure 1c. In contrast, although interactions potentially lift degeneracies, in non-SOC BLG, K+−K− degeneracy yields two doubly degenerate LLs in each quartet so that the gaps vanish at D = 0 7031
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Figure 4. Corrections to the single particle model due to Coulomb interactions. (a) Schematic representation of the broken symmetry gaps for N = 2 at D = 0. ΔK and ΔK′ stands for the proximity induced Ising and Rashba SOC, causing different splitting in the K+ and K− valleys. JK+ and JK− are the Coulomb interactions for the electrons with opposite spins in the same valley. (b) Extracted differential ΔJK− relative to its value at B = 6 T at D = 0.
λR = 22 meV. However, these crossing points are in principle still potentially affected by Coulomb interactions. To elucidate the role of Coulomb interactions in the behavior of the system, we determine the transport gaps by measuring the temperature dependence of Rxx in a number of valleys versus D for device D1. An example of this behavior is shown in Figure 3a which shows Arrhenius plots for ν = 6 at D = 0 versus B. We then fit the Rxx minima against 1/T with the ÅÄÅ Δ − Γ ÑÉÑ equation R = R 0 expÅÅÅ− 2k T ÑÑÑ to extract the quantity Δ − Γ, ÅÇ B Ñ Ö where Δ is the gap and Γ (full width at half-maximum) is the LL width. Figure 3b shows the extracted Δ − Γ versus B for ν = 5, 6, and 7. To make a comparison to the single particle model above, we consider the rate of change of Δ − Γ with D [i.e., d(Δ − Γ)/dD = dΔ/dD assuming constant Γ] near a gap closing, such as shown in Figure 3c. This is insensitive to Γ, as well as Coulomb interactions within a Hartree−Fock picture. Table 1 in Figure 3 shows the measured values for a number of crossings. Values on both sides of the crossings are averaged if both are obtained. We then compute the expected dΔ/dD at each crossing and using only λR as a free parameter minimize the sum of the squared differences between all the expected and measured crossing rates with respect to λR. We find a best fit value of λR ≈ 15 ± 5 meV for the data shown. The computed values using this λR value are tabulated in Figure 3 Table 1 column 4 which show reasonable agreement to the measured values. However, deviations exist which may, for example, stem from measurement uncertainty in the gap closing rates. This indicates that the λR obtained from this method should be considered only an estimate and characteristically at ∼15 meV, consistent with the range of previously found values, for example, refs 9 and 15. Using this extracted value of λR, we compute the expected behavior of the ν = 5 gap and plot it as the dashed line in Figure 3b without free parameters. Fitting a straight line to the ν = 5 gap data in Figure 3b yields a slope of 0.11 ± 0.02 meV/ T, very close to the theoretical slope of ∼gμB = 0.115 meV/T, indicating that Coulomb interactions have only a small effect on the gap. The ν = 7 gap is slightly larger but similar in magnitude. On the other hand, the fitted slope of 0.245 meV/ T of the ν = 6 gap is significantly larger than gμB and also
For λ > EZ, the ν = ±3 crossing states therefore have the same orbital but opposite spins (see SI Figure S4). The crossing points are B-dependent as found in ref 22 (see also Figure S5). In this case, the ν = ±3 crossing points both occur at D = 0 when λ = 2EZ and then switch their signs of D for larger B.22 In contrast, for λ < EZ (including negative λ values) the ν = ±3 crossing states have the same orbital and spin (see SI Figure S4). The crossing points are thus B-independent. (We note that in the zero-energy LLs Rashba coupling is expected to contribute negligibly small level shifts ∼λR3/γ12, where γ1 = 0.361 eV.20 An exception to this can occur at a potential level crossing within the same valley in which case the Rashba term can produce an anticrossing at a noticeably larger energy scale. However, this does not affect the ν = ±3 crossings.) The observed B-insensitivity of the ν = ±3 crossings thus implies that λ < EZ in device D1. By carefully comparing the ν = ±3 crossing D values with the theory, we find λ = −2.2 ± 0.1 meV. Note that at the maximum B measured, B = 10 T, EZ ≈ 1.1 meV, thus if λ were positive but of the same magnitude we would have instead λ > EZ. This would lead to a strong B dependence of the ν = ±3 crossing points reaching D = 0 when EZ = λ/2 = 1.1 meV corresponding to B ≈ 10 T, contradicting our observations. A negative λ is also indicated by the crossings for ν = 3 and 5 having the same sign of D (see SI Figure S4) at B = 8 T. This is consistent with theoretical predictions that negative λ occurs for certain ranges of graphene−WSe2 twist angles.30 (Similar results were also obtained from another device D2, shown in SI Figure S6.) For the Rashba coupling, expected to be positive regardless of twist angle,30 we measure the crossing points’ D-values for ν = ±5, ±6, and ±7. Figure 2c plots this data for B = 6 T and Figure 2d plots the corresponding data for B = 10 T. Using λ = −2.2 meV as obtained above and using λR as the only free parameter, we minimize the sum of all the squared deviations between the predicted single particle values and the measured data, obtaining λR ∼ 20 ± 1 meV. Figure 2c,d shows the resulting model values using the best fit λR plotted as open triangles. The single particle model with this value of λR yields a reasonable agreement to the measured values. Device D2 (SI Figure S6) also showed similar results with λ = −2.6 meV and 7032
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increases with B. In a single particle picture, the quartet of LLs associated with these gaps consists of states that can be labeled by a K+ or K− valley index and spin. The K+ and K− states are split by the SOC into nearly up and down spin states (with slight canting), which are oppositely directed in the two valleys, effectively. The single particle calculation shows that although the Rashba SOC reduces the total splitting, the ν = 5 gap varies linearly with B with a slope ≈ gμΒ, as observed. The measured energy is lower than the predicted single particle splitting, which we attribute to an LL width of ∼0.4 meV. The results of the single particle model for all three gaps ν = 5, 6, and 7 are plotted over a wider range of B in the Figure 3b inset. Although this picture produces reasonable results over the measured B range for Δ5 and Δ7, it predicts that Δ6 starts at a splitting set by the SOC and decreases with B, contrary to what is observed. We therefore consider a model that includes Coulomb interactions. SOC breaks the K+-K− symmetry, leading to potentially different exchange energies in the two valleys. Additionally, there could be an exchange coupling between the K+ and K− states. However, based on the close agreement in the behavior of the Δ5 and Δ7 gaps to the single particle picture we assume this is negligible. Thus, we model the Coulomb interaction with two parameters, which are JK+ and JK−, corresponding to the exchange energy to flip a spin within an LL in the K+ or K− valleys with an effective single particle picture shown schematically in Figure 4a. The similar energy scales measured for Δ5 and Δ7 indicate that JK+ ≈ JK−. We obtain JK− by taking the total gap Δ6 to be Δ6−sp + JK−, where Δ6−sp is the expected single particle gap at D = 0, plotted in the inset to Figure 4b, and rearranging it to find JK− = Δ6 − Δ6−sp. Because of the unknown Coulomb interactions outside of the measured range and the uncertainty in the LL width, this quantity is plotted in Figure 4b as a differential ΔJK− relative to its value at B = 6 T. The behavior is approximately linear in B. Although it may be initially expected that JK− is proportional to the Coulomb energy scale e2/εlB, where lB is the magnetic length and ε is the dielectric constant, this result is in agreement with previous experiments that also find an approximately linear in B Coulomb interaction.31,32 This may arise from changes to the dielectric constant with B.42 However, more work will be required to understand the origin of this behavior. In conclusion, we have measured high mobility bilayer graphene samples in which the spin−orbit interaction is induced by proximity coupling to WSe2. The Ising parameter is measured to be λ ≈ −2.2 meV, whereas the Rashba parameter is ∼15 meV. The SOC breaks all degeneracies of the LLs at D = 0, and although the crossing points versus D are qualitatively in agreement with a single particle model both the gaps and the crossing points can be affected by Coulomb interactions. We measure the Coulomb energy corrections for ν = 6 and find it is approximately linear in B over the measured range, in accordance with previous results on non-SOC bilayer graphene. The quantum Hall effect is a useful probe of SOC in bilayer graphene. In future work, this approach may be used to test theories in which both the Ising and Rashba SOC are tunable by the twist angle between the graphene and WSe2,30,43 potentially affording a unique approach to controlling SOC in materials.
Letter
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.9b02445.
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Additional discussion of single particle model and data analysis as well as additional figures (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Phone: 614-292-0375. ORCID
Dongying Wang: 0000-0002-9674-3532 Chun Ning Lau: 0000-0003-2159-6723 Marc Bockrath: 0000-0002-7000-1442 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Jyoti Katoch for helpful discussions. This research funded by DOE ER 46940-DE-SC0010597. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan, A3 Foresight by JSPS, and the CREST(JPMJCR15F3), JST.
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